The study examines the regularity of spheres in homogeneous groups using specific distance criteria.
problem Understanding the regularity of metric spheres in homogeneous groups.
method Investigation of left-invariant distances on Lie groups with homothetic automorphisms, focusing on Carnot groups and Heisenberg group.
result Established criteria for the regularity of metric spheres in homogeneous groups, including the Heisenberg group.
A manifold M is locally conformally Kahler (LCK) if it admits a Kahler covering with monodromy acting by holomorphic homotheties. Let M be an LCK manifold admitting a holomorphic conformal flow of diffeomorphisms, lifted to a non-isometric homothetic flow on its covering. We show that M admits an automorphic pote…
New rigidity result for hyperbolic surfaces based on curve lengths.
problem Determining hyperbolic metrics on surfaces from curve lengths.
method Investigating oriented graphs on curve complexes and Dehn quasi-homothetic functions.
result Knowing which curve is longer suffices to determine the hyperbolic metric on a surface.
We prove the following results: (i) A Sasakian metric as a non-trivial Ricci soliton is null η-Einstein, and expanding. Such a characterization permits to identify the Sasakian metric on the Heisenberg group H2n+1 as an explicit example of (non-trivial) Ricci soliton of such type. (ii) If an η-Einste…
We show that the horoboundary of outer space for the Lipschitz metric is a quotient of Culler and Morgan's classical boundary, two trees being identified whenever their translation length functions are homothetic in restriction to the set of primitive elements of FN. We identify the set of Busemann points with the s…
The paper proves no homothetical surfaces exist in 3D pseudo-Galilean space.
problem Exploring homothetical surfaces in pseudo-Galilean space.
method Analyzing surfaces defined by a function and satisfying specific Laplacian conditions.
result No homothetical surfaces exist in 3D pseudo-Galilean space.
Proves uniqueness of catenoid-like shapes in a ball.
problem Uniqueness of catenoid-like minimal surfaces.
method Analyzes σ-homothetic free boundary minimal annuli. result Critical catenoid is the only σ-homothetic shape. Characterizes and describes selfsimilar Hessian manifolds with homothetic vector fields.
problem Understanding the structure and properties of selfsimilar Hessian manifolds.
method Characterization and description of selfsimilar manifolds with homothetic vector fields.
result Any selfsimilar Hessian manifold with a potential homothetic vector field is locally isomorphic to a product of radiant Hessian manifolds.
The paper explains geometric correspondences for homothetic navigation.
problem Understanding geodesic and Jacobi field correspondences in homothetic navigation.
method Providing conceptual explanations and shortcuts to formulas.
result Directly seeing local correspondence between isoparametric functions or hypersurfaces.
Solves geodesic completeness on pseudo-homothetic Lie group.
problem Geodesic completeness on pseudo-homothetic Lie group.
method Exhibited a family of complete metrics with bounded geodesic velocity.
result Set of complete metrics is not closed.
In this paper we introduce entropy-stability and F-stability for homothetically shrinking Yang-Mills solitons, employing entropy and second variation of F-functional respectively. For a homothetically shrinking soliton which does not descend, we prove that entropy-stability implies F-stability. These stabil…
Mathematically, a homothetic function is a function of the form f(x)=F(h(x1,...,xn)), where h is a homogeneous function of any degree d=0 and F is a monotonically increasing function. In economics homothetic functions are production functions whose marginal technical rate of substitution is homogeneo…
Paper classifies ruled surfaces in Lorentz-Minkowski space for a specific flow.
problem Classifying ruled surfaces in Lorentz-Minkowski space.
method Examining homothetic self-similar solutions of the inverse mean curvature flow.
result Existence of two classes of non-cylindrical homothetic solitons.
Paper studies conformal vector fields in specific Finsler spaces.
problem Characterizing and understanding conformal vector fields in (α,β) spaces. method Using principles to characterize and determine local structure of conformal vector fields under curvature conditions.
result Construction of non-homothetic conformal vector fields on locally projectively Randers spaces.
The paper explores conditions for homothetic Killing vectors on spacetime hypersurfaces.
problem Conditions for the existence of homothetic Killing vectors on spacetime hypersurfaces.
method General identities relating deformation tensor and tensor on hypersurfaces, applied to specific settings.
result Necessary and sufficient conditions for homothetic Killing vectors on spacetime hypersurfaces.
We show that any horizontally homothetic submersion from a compact manifold of nonnegative sectional curvature is a Riemannian submersion.
In \cite{Mul} one-parameter planar motion was first introduced and the relations between absolute, relative, sliding velocities (and accelerations) in the Euclidean plane E2 were obtained. Moreover, the relations between the Complex velocities one-parameter motion in the Complex plane were provided by \cite…
The paper represents performance processes in incomplete markets using BSDE.
problem Incomplete markets with stochastic factors.
method Ergodic and infinite horizon BSDEs for homothetic forward performance processes.
result Derivation of representations for power, exponential, and logarithmic forward performance processes.
The aim of this paper is to classify compact, simply connected Kähler manifolds which admit totally geodesic, holomorphic complex homothetic foliation by curves.
In this article we derive a complete classification of all submanifolds in space forms with codimension two for which the Gauss map is homothetic.
We obtain a local classification of complex homothetic foliations on Kaehler manifolds by complex curves. This is used to construct almost Kaehler, Ricci-flat metrics subject to additional curvature properties.
We classify homothetical surfaces with constant mean curvature in hyperbolic space.
problem Classifying surfaces with constant mean curvature in hyperbolic space.
method Using the upper half-space model, we define surfaces by z=φ(x)ψ(y) and prove they are parabolic. result All homothetical surfaces with constant mean curvature in hyperbolic space are parabolic.
Study of homothetic solitons in inverse mean curvature flow.
problem Understanding the behavior of solitons in inverse mean curvature flow.
method Analyzing solutions that evolve by homotheties of a given submanifold.
result Classification of rotationally invariant Lagrangian homothetic solitons.
Study on minimal surfaces in a 3D space with 2m-norm.
problem Characterizing minimal surfaces in a specific geometric space.
method Examining translation, homothetical, and separable minimal surfaces.
result New insights into minimal surfaces in a 3D space with 2m-norm.
Study on solitons in deformed Kenmotsu manifolds with specific vector fields.
problem Analyzing geometric solitons in deformed Kenmotsu manifolds.
method Examined almost Riemann and Ricci solitons in a D-homothetically deformed Kenmotsu manifold with specific vector fields. result Explicitly obtained Ricci and scalar curvatures for some cases, provided a lower bound for Ricci curvature.
The paper classifies vertices in planar polygons formed by convex domains.
problem Classifying vertices in planar polygons formed by convex domains.
method Analyzing polygons formed by homothets and translates of a convex domain.
result The number of singular boundary points in a C-polygon is between n and 2(n−1)+m for a strictly convex domain with m singular boundary points. The paper classifies hypersurfaces with specific curvature for economic applications.
problem Classifying hypersurfaces with null Gauss-Kronocker curvature.
method Complete classification of homothetical hypersurfaces in Euclidean space.
result Applications to production functions in economics.
Existence of unstable shrinking solutions in fractional mean curvature flow.
problem Existence and stability of self-shrinkers in fractional mean curvature flow.
method Existence proof of homothetically shrinking solutions with prescribed boundary conditions.
result Unstable shrinking solutions, except the ball, for fractional mean curvature flow.
We consider graphical solutions to mean curvature flow and obtain a stability result for homothetically expanding solutions coming out of cones of positive mean curvature: If another solution is initially close to the cone at infinity, then the difference to the homothetically expanding solution becomes small for large…
Flat holonomies imply homotheticity in certain curved spaces.
problem Characterizing manifolds with specific holonomy properties.
method Analyzing horospheres and holonomy in curved spaces.
result Closed, strictly pinched negatively curved manifolds with matching holonomies are homothetic to hyperbolic spaces.
In this paper, it is proved that any conformal vector field is homothetic on a locally projectively flat (α,β)-space of non-Randers type in dimension n≥3, and the local solutions of such a vector field are determined. While on a locally projectively flat Randers space, examples showthat the conformal vector fiel…
This paper studies rapidly forming singularities in the Yang-Mills flow. It is shown that a sequence of blow-ups near the singular point converges, modulo the gauge group, to a homothetically shrinking soliton with non-zero curvature. The proof uses Hamilton's monotonicity formula. Examples of homothetically shrinking …
Proves stability of cone-volume measure with nearly constant density.
problem Stability of cone-volume measure with near constant density.
method Proves stability of cone-volume measure with near constant density.
result Homothetic copy of the body is close to the unit ball in the L2-distance. Symmetry groups help define solitons in curved spaces.
problem Understanding solitons in curved spaces.
method Defined generalized solitons using symmetry groups.
result Affine solutions are self-similar.
In the study of the curve shortening flow on general closed curves, Abresch and Langer posed a conjecture that the homothetic curves can be regarded as saddle points between multi-folded circles and some singular curves. In other words, these homothetic curves are the watershed between curves with a nonsingular future …
Study of evolutes of polygons and curves in higher dimensions.
problem Understanding evolutes of spatial polygons and curves in higher dimensions.
method Analyzing iterations of evolute transformations and studying properties of evolutes for polygons and curves.
result Eigenvalues of the second evolute map have double multiplicity, and evolutes of certain curves are homothetic to the curves themselves.
The paper analyzes self-similar solutions for mean curvature flow in 3D.
problem Analyzing self-similar solutions for mean curvature flow in R3. method Analysis of self-similar solutions for surfaces of revolution, ruled surfaces, and cylindrical surfaces under homothetic helicoidal motions.
result Characterization and explicit families of exact solutions for cylindrical surfaces.
In this paper we study the set of balanced metrics (in Donaldson's terminology) on a compact complex manifold M which are homothetic to a given balanced one. This question is related to various properties of the Tian-Yau-Zelditch approximation theorem for Kahler metrics. We prove that this set is finite when M admits…
In this paper we extend some well-known rigidity results for conformal changes of Einstein metrics to the class of generalized quasi-Einstein (GQE) metrics, which includes gradient Ricci solitons. In order to do so, we introduce the notions of conformal diffeomorphisms and vector fields that preserve a GQE structure. W…
The flow of curves in Minkowski plane converges to a specific shape.
problem Analyzing curve diffusion in Minkowski plane.
method Anisotropic polyharmonic curve flow.
result Closed curves converge to a homothetic rescaling of the isoperimetrix.
Study on CPSRM from/to Kähler manifolds, deriving integrability and geodesic results.
problem Existence and properties of CPSRM from/to Kähler manifolds.
method Analytical derivation of properties, examples, and conditions for homotheticity and harmonicity.
result Derived integrability and geodesic conditions for CPSRM.
Kähler cones over Sasakian manifolds are flat if projectively induced.
problem Characterizing Kähler cones over Sasakian manifolds.
method Relating Kähler potentials and using Ricci-flatness.
result Kähler cones over regular Sasakian manifolds are flat if projectively induced.
The study examines properties of quasi-Para-Sasakian manifolds and their curvature.
problem Investigating curvature properties of quasi-Para-Sasakian manifolds.
method Basic properties and general curvature identities of quasi-Para-Sasakian manifolds are derived.
result If a quasi-Para-Sasakian manifold has constant curvature, it must be non-positive, and under specific conditions, it can be paracosymplectic or obtained by a homothetic deformation of a para-Sasakian structure.
New proof of non-compact homothetic solitons for inverse mean curvature flow.
problem Existence of non-compact homothetic solitons for inverse mean curvature flow.
method Analytical proof using differential equations.
result Existence of a unique solution for the given equation.
Constructs homogeneous Kähler structures on tangent bundles of Hessian manifolds.
problem Creating Kähler structures on tangent bundles of Hessian manifolds.
method Endowing Hessian manifolds with Kähler structures using group actions and homothetic vector fields.
result Homogeneous conformally Kähler structures on tangent bundles of selfsimilar Hessian manifolds.
Study on Kähler-Ricci flow and conformal submersion singularity formation.
problem Singularity formation of Kähler-Ricci flow on manifolds with conformal submersion.
method Derive conditions for the preservation of conformal submersion and analyze singularity formation.
result Formation of type I singularity and standard splitting of Cheeger-Gromov limit.
The study introduces a new soliton concept to classify Sasakian 3-manifolds.
problem Classifying Sasakian 3-manifolds under specific conditions.
method Introducing and studying ∗-Ricci-Yamabe solitons on contact metric manifolds. result Sasakian 3-manifolds admitting ∗-Ricci-Yamabe solitons are ∗-Ricci flat, positive Sasakian, and have Fano transverse geometry. The canonical paracontact connection is defined and it is shown that its torsion is the obstruction the paracontact manifold to be paraSasakian. A D-homothetic transformation is determined as a special gauge transformation. The η-Einstein manifold are defined, it is prove that their scalar curvature is a …